Theorem mobii 2063

Description: Formula-building rule for "at most one" quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)

Hypothesis

Ref	Expression
mobii.1	⊢ (𝜓 ↔ 𝜒)

Assertion

Ref	Expression
mobii	⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)

Proof of Theorem mobii

Step	Hyp	Ref	Expression
1		mobii.1	. . . 4 ⊢ (𝜓 ↔ 𝜒)
2	1	a1i 9	. . 3 ⊢ (⊤ → (𝜓 ↔ 𝜒))
3	2	mobidv 2062	. 2 ⊢ (⊤ → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
4	3	mptru 1362	1 ⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)

Syntax hints:   ↔ wb 105  ⊤wtru 1354  ∃*wmo 2027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-eu 2029  df-mo 2030

This theorem is referenced by:  moaneu  2102  moanmo  2103  2moswapdc  2116  2exeu  2118  rmobiia  2667  rmov  2758  euxfr2dc  2923  rmoan  2938  2rmorex  2944  mosn  3629  dffun9  5246  funopab  5252  funco  5257  funcnv2  5277  funcnv  5278  fun2cnv  5281  fncnv  5283  imadif  5297  fnres  5333  ovi3  6011  oprabex3  6130  axaddf  7867  axmulf  7868  frecuzrdgtcl  10412  frecuzrdgfunlem  10419  fsum3  11395